undecidability in group theory, topology, and...

Undecidability ingroup theory,topology, and

analysis

Bjorn Poonen

Group theory

F.p. groups

Word problem

Markov properties

Topology

Fundamental group

Homeomorphismproblem

Manifold?

Knot theory

Analysis

Inequalities

Complex analysis

Integration

Undecidability in group theory, topology, andanalysis

Bjorn Poonen

Rademacher Lecture 2November 7, 2017


analysis

Bjorn Poonen

Group theory

F.p. groups

Word problem

Markov properties

Topology

Fundamental group


Manifold?

Knot theory

Analysis

Inequalities

Complex analysis

Integration

Group theory

Question

Can a computer decide whethertwo given elements of a group are equal?

To make sense of this question, we must specify

1. how the group is described

2. how the element is described

The descriptions should be suitable for input into a Turingmachine.


analysis

Bjorn Poonen

Group theory

F.p. groups

Word problem

Markov properties

Topology

Fundamental group


Manifold?

Knot theory

Analysis

Inequalities

Complex analysis

Integration

Group theory

Question

Can a computer decide whethertwo given elements of a group are equala given element of a group equals the identity?


1. how the group is described

2. how the element is described



analysis

Bjorn Poonen

Group theory

F.p. groups

Word problem

Markov properties

Topology

Fundamental group


Manifold?

Knot theory

Analysis

Inequalities

Complex analysis

Integration

Group theory

Question

Can a computer decide whethertwo given elements of a group are equala given element of a group equals the identity?


1. how the group is described: f.p. group

2. how the element is described: word



analysis

Bjorn Poonen

Group theory

F.p. groups

Word problem

Markov properties

Topology

Fundamental group


Manifold?

Knot theory

Analysis

Inequalities

Complex analysis

Integration

Example: The symmetric group S3

In cycle notation, r = (123) and t = (12). These satisfy

r3 = 1, t2 = 1, trt−1 = r−1

It turns out that r and t generate S3, and every relationinvolving them is a consequence of the relations above:

S3 = 〈r , t | r3 = 1, t2 = 1, trt−1 = r−1〉.


analysis

Bjorn Poonen

Group theory

F.p. groups

Word problem

Markov properties

Topology

Fundamental group


Manifold?

Knot theory

Analysis

Inequalities

Complex analysis

Integration

Finitely presented groups

Definition

An f.p. group is a group specified by finitely manygenerators and finitely many relations.

Example

Z× Z = 〈a, b | ab = ba〉

Example

The free group on 2 (noncommuting) generators is

F2 := 〈a, b | 〉


analysis

Bjorn Poonen

Group theory

F.p. groups

Word problem

Markov properties

Topology

Fundamental group


Manifold?

Knot theory

Analysis

Inequalities

Complex analysis

Integration

Representing elements of an f.p. group: words

S3 = 〈r , t | r3 = 1, t2 = 1, trt−1 = r−1〉.

Definition

A word is a sequence of the generator symbols and theirinverses, such as

tr−1ttrt−1rrr .

Since r and t generate S3, every element of S3 is representedby a word, but not necessarily in a unique way.

Example

The words tr and r−1t both represent (23).


analysis

Bjorn Poonen

Group theory

F.p. groups

Word problem

Markov properties

Topology

Fundamental group


Manifold?

Knot theory

Analysis

Inequalities

Complex analysis

Integration

The word problem

Given an f.p. group G , we have

Word problem for G

Find an algorithm with

input: a word w in the generators of Goutput: YES or NO, according to whether w = 1 in G .

Harder problem:

Uniform word problem

Find an algorithm with

input: an f.p. group G , and a word w in thegenerators of G

output: YES or NO, according to whether w = 1 in G .


analysis

Bjorn Poonen

Group theory

F.p. groups

Word problem

Markov properties

Topology

Fundamental group


Manifold?

Knot theory

Analysis

Inequalities

Complex analysis

Integration

Word problem for Fn

Theorem

The word problem for the free group Fn is decidable.

Algorithm to decide whether a given word w represents 1:

1. Repeatedly cancel adjacent inverses until there isnothing left to cancel.

2. Check if the end result is the empty word.

Example

In the free group F2 = 〈a, b〉, given the word

aba−1bb−1abb,

cancellation leads toabbb,

which is not the empty word,so aba−1bb−1abb does not represent the identity.


analysis

Bjorn Poonen

Group theory

F.p. groups

Word problem

Markov properties

Topology

Fundamental group


Manifold?

Knot theory

Analysis

Inequalities

Complex analysis

Integration

Undecidability of the word problem

Theorem (P. S. Novikov and Boone,independently in the 1950s)

There exists an f.p. group G such that the word problem forG is undecidable.

The strategy of the proof, as for Hilbert’s tenth problem, isto build a group G such that solving the word problem for Gis at least as hard as solving the halting problem.

Corollary

The uniform word problem is undecidable.


analysis

Bjorn Poonen

Group theory

F.p. groups

Word problem

Markov properties

Topology

Fundamental group


Manifold?

Knot theory

Analysis

Inequalities

Complex analysis

Integration

Markov properties

Definition

A property of f.p. groups is called a Markov property if

1. there exists an f.p. group G1 with the property, and

2. there exists an f.p. group G2 that cannot be embeddedin any f.p. group with the property.

Example

The property of being finite is a Markov property, because

1. There exists a finite group!

2. Z cannot be embedded in any finite group.

Other Markov properties: trivial, abelian, free, . . . .


analysis

Bjorn Poonen

Group theory

F.p. groups

Word problem

Markov properties

Topology

Fundamental group


Manifold?

Knot theory

Analysis

Inequalities

Complex analysis

Integration

Theorem (Adian & Rabin 1955–1958)

For each Markov property P, the problem of decidingwhether an arbitrary f.p. group has P is undecidable.

Sketch of proof.

Embed the uniform word problem in this P problem:Given an f.p. group G and a word w in its generators,build another f.p. group K such that

K has P ⇐⇒ w = 1 in G .

Example

There is no algorithm to decide whether an f.p. group istrivial.


analysis

Bjorn Poonen

Group theory

F.p. groups

Word problem

Markov properties

Topology

Fundamental group


Manifold?

Knot theory

Analysis

Inequalities

Complex analysis

Integration

Fundamental groupFix a manifold M.

and a point p.Consider paths in M that start and end at p.Paths are homotopic if one can be deformed to the other.

Fundamental group π1(M) := {paths}/homotopy.

Group law: concatenation of paths.


analysis

Bjorn Poonen

Group theory

F.p. groups

Word problem

Markov properties

Topology

Fundamental group


Manifold?

Knot theory

Analysis

Inequalities

Complex analysis

Integration

Fundamental groupFix a manifold M and a point p.

Consider paths in M that start and end at p.Paths are homotopic if one can be deformed to the other.




analysis

Bjorn Poonen

Group theory

F.p. groups

Word problem

Markov properties

Topology

Fundamental group


Manifold?

Knot theory

Analysis

Inequalities

Complex analysis

Integration

Fundamental groupFix a manifold M and a point p.Consider paths in M that start and end at p.

Paths are homotopic if one can be deformed to the other.




analysis

Bjorn Poonen

Group theory

F.p. groups

Word problem

Markov properties

Topology

Fundamental group


Manifold?

Knot theory

Analysis

Inequalities

Complex analysis

Integration

Fundamental groupFix a manifold M and a point p.Consider paths in M that start and end at p.Paths are homotopic if one can be deformed to the other.




analysis

Bjorn Poonen

Group theory

F.p. groups

Word problem

Markov properties

Topology

Fundamental group


Manifold?

Knot theory

Analysis

Inequalities

Complex analysis

Integration

Examples of fundamental groups

π1(torus) = Z× Z π1(sphere) = {1}

This gives one way to prove that the torus and the sphereare not homeomorphic, i.e., that they do not have the sameshape even after stretching.


analysis

Bjorn Poonen

Group theory

F.p. groups

Word problem

Markov properties

Topology

Fundamental group


Manifold?

Knot theory

Analysis

Inequalities

Complex analysis

Integration

The homeomorphism problem

Question

Given two manifolds, can one decide whether they arehomeomorphic?

To make sense of this question, we must specify how amanifold is described.

This will be done using the notion of simplicial complex.


analysis

Bjorn Poonen

Group theory

F.p. groups

Word problem

Markov properties

Topology

Fundamental group


Manifold?

Knot theory

Analysis

Inequalities

Complex analysis

Integration

Simplicial complexes

Definition

Roughly speaking, a finite simplicial complex is a finite unionof simplices (points, segments, triangles, tetrahedra, . . . )together with data on how they are glued. The description ispurely combinatorial.

Example

The icosahedron is a finite simplicial complexhomeomorphic to the 2-sphere S2.

From now on, manifold means “compact manifoldrepresented by a particular finite simplicial complex”,so that it can be the input to a Turing machine.


analysis

Bjorn Poonen

Group theory

F.p. groups

Word problem

Markov properties

Topology

Fundamental group


Manifold?

Knot theory

Analysis

Inequalities

Complex analysis

Integration

Undecidability of the homeomorphism problem

Theorem (Markov 1958)

The problem of deciding whether two manifolds arehomeomorphic is undecidable.

Sketch of proof.

Let n ≥ 5. Given an f.p. group G and a word w in itsgenerators, one can construct a n-manifold ΣG ,w such that

1. If w = 1 in G , then ΣG ,w ≈ Sn.

2. If w 6= 1, then π1(ΣG ,w ) is nontrivial (so ΣG ,w 6≈ Sn).

Thus, if the homeomorphism problem were decidable, thenthe uniform word problem would be too. But it isn’t.

In fact, the homeomorphism problem is known to be

decidable in dimensions ≤ 3, and

undecidable in dimensions ≥ 4.


analysis

Bjorn Poonen

Group theory

F.p. groups

Word problem

Markov properties

Topology

Fundamental group


Manifold?

Knot theory

Analysis

Inequalities

Complex analysis

Integration

The previous proof showed that for n ≥ 5, the manifold Sn isunrecognizable: the problem of deciding whether a givenn-manifold is homeomorphic to Sn is undecidable.

Theorem (S. P. Novikov 1974)

Each n-manifold M with n ≥ 5 is unrecognizable.

Question

Is S4 recognizable? (The answer is not known.)

To explain the idea of the proof of the theorem, we need thenotion of connected sum.


analysis

Bjorn Poonen

Group theory

F.p. groups

Word problem

Markov properties

Topology

Fundamental group


Manifold?

Knot theory

Analysis

Inequalities

Complex analysis

Integration

Connected sumThe connected sum of n-manifolds M and N is then-manifold obtained by cutting a small disk out of each andconnecting them with a tube.


analysis

Bjorn Poonen

Group theory

F.p. groups

Word problem

Markov properties

Topology

Fundamental group


Manifold?

Knot theory

Analysis

Inequalities

Complex analysis

Integration


analysis

Bjorn Poonen

Group theory

F.p. groups

Word problem

Markov properties

Topology

Fundamental group


Manifold?

Knot theory

Analysis

Inequalities

Complex analysis

Integration

Am I a manifold?

Theorem

It is impossible to decide whether a finite simplicial complexis homeomorphic to a manifold.

Proof.

SΣG ,w := suspension over our possibly fake sphere ΣG ,w .

If w = 1 in G , then ΣG ,w ≈ Sn, so SΣG ,w ≈ Sn+1.

If w 6= 1, then SΣG ,w is not locally Euclidean.


analysis

Bjorn Poonen

Group theory

F.p. groups

Word problem

Markov properties

Topology

Fundamental group


Manifold?

Knot theory

Analysis

Inequalities

Complex analysis

Integration

Knot theory

Definition

A knot is an embedding of the circle S1 in R3.

Definition

Two knots are equivalent if one can be deformed into theother within R3, without crossing itself.


analysis

Bjorn Poonen

Group theory

F.p. groups

Word problem

Markov properties

Topology

Fundamental group


Manifold?

Knot theory

Analysis

Inequalities

Complex analysis

Integration

From now on, knot means “a knot obtained by connecting afinite sequence of points in Q3”, so that it admits a finitedescription.

Theorem (Haken 1961 and Hemion 1979)

There is an algorithm that takes as input two knots in R3

and decides whether they are equivalent.


analysis

Bjorn Poonen

Group theory

F.p. groups

Word problem

Markov properties

Topology

Fundamental group


Manifold?

Knot theory

Analysis

Inequalities

Complex analysis

Integration

Higher-dimensional knots

Though the knot equivalence problem is decidable, ahigher-dimensional analogue is not:

Theorem (Nabutovsky & Weinberger 1996)

If n ≥ 3, the problem of deciding whether two embeddings ofSn in Rn+2 are equivalent is undecidable.

Question

What about n = 2? Not known.


analysis

Bjorn Poonen

Group theory

F.p. groups

Word problem

Markov properties

Topology

Fundamental group


Manifold?

Knot theory

Analysis

Inequalities

Complex analysis

Integration

Polynomial inequalities

Question

Which of the following inequalities are true for all real valuesof the variables?

a2 + b2 ≥ 2ab

TRUE

x4 − 4x + 5 ≥ 0

TRUE

536x287196896 − 210y287196896 + 777x3y16z4732987

−1111x54987896 − 2823y927396 + 27x94572y9927z999

−936718x726896 + 887236y726896 − 9x24572y7827z13

+89790876x26896 + 30y26896 + 987x245y6z6876

+9823709709790790x28 − 1987y28 + 1467890461986x2y6z4

+80398600x2z12 − 27980186xy + 3789720156y2 + 9328769x

−1956820y − 275893249827098790768645846898z ≥ −389?

FALSE


analysis

Bjorn Poonen

Group theory

F.p. groups

Word problem

Markov properties

Topology

Fundamental group


Manifold?

Knot theory

Analysis

Inequalities

Complex analysis

Integration


Question


a2 + b2 ≥ 2ab TRUE

x4 − 4x + 5 ≥ 0

TRUE

536x287196896 − 210y287196896 + 777x3y16z4732987

−1111x54987896 − 2823y927396 + 27x94572y9927z999

−936718x726896 + 887236y726896 − 9x24572y7827z13

+89790876x26896 + 30y26896 + 987x245y6z6876

+9823709709790790x28 − 1987y28 + 1467890461986x2y6z4

+80398600x2z12 − 27980186xy + 3789720156y2 + 9328769x

−1956820y − 275893249827098790768645846898z ≥ −389?

FALSE


analysis

Bjorn Poonen

Group theory

F.p. groups

Word problem

Markov properties

Topology

Fundamental group


Manifold?

Knot theory

Analysis

Inequalities

Complex analysis

Integration


Question


a2 + b2 ≥ 2ab TRUE

x4 − 4x + 5 ≥ 0 TRUE

536x287196896 − 210y287196896 + 777x3y16z4732987

−1111x54987896 − 2823y927396 + 27x94572y9927z999

−936718x726896 + 887236y726896 − 9x24572y7827z13

+89790876x26896 + 30y26896 + 987x245y6z6876

+9823709709790790x28 − 1987y28 + 1467890461986x2y6z4

+80398600x2z12 − 27980186xy + 3789720156y2 + 9328769x

−1956820y − 275893249827098790768645846898z ≥ −389?

FALSE


analysis

Bjorn Poonen

Group theory

F.p. groups

Word problem

Markov properties

Topology

Fundamental group


Manifold?

Knot theory

Analysis

Inequalities

Complex analysis

Integration

Polynomial inequalities, continued

Question

Can a computer decide, given a polynomial inequality

f (x1, . . . , xn) ≥ 0

with rational coefficients, whether it is true for all realnumbers x1, . . . , xn?

YES! (Tarski 1951) More generally, it can decide the truth ofany first-order sentence involving polynomial inequalities.

How? For example, how could it decide whether a given setdefined by a Boolean combination of inequalities is empty?


analysis

Bjorn Poonen

Group theory

F.p. groups

Word problem

Markov properties

Topology

Fundamental group


Manifold?

Knot theory

Analysis

Inequalities

Complex analysis

Integration

Inequalities: induction on the number of variables

x2 + y2 < 1

x > −1 ∧ x < 1

In general, the projection (x1, . . . , xn) 7→ (x1, . . . , xn−1)maps a set S defined by an explicit Booleancombination of inequalities to another such set S ′.

S 6= ∅ if and only if S ′ 6= ∅.Keep projecting until only 1 variable is left; then usecalculus.


analysis

Bjorn Poonen

Group theory

F.p. groups

Word problem

Markov properties

Topology

Fundamental group


Manifold?

Knot theory

Analysis

Inequalities

Complex analysis

Integration

Exponential inequalities

Can a computer decide the truth of inequalities like

eex+y

+ 20 ≥ 5x + 4y ?

Warmup: What about ee3/2

+ e5/3 ≥ 13396

143?

This should be easy: compute both sides to high precision,but. . .

What if they turn out to be exactly equal?

Schanuel’s conjecture in transcendental number theorypredicts that “coincidences” like these never occur, but ithas not been proved.

Theorem (Macintyre and Wilkie)

If Schanuel’s conjecture is true, then exponential inequalitiesin any number of variables are decidable.


analysis

Bjorn Poonen

Group theory

F.p. groups

Word problem

Markov properties

Topology

Fundamental group


Manifold?

Knot theory

Analysis

Inequalities

Complex analysis

Integration

Trigonometric inequalities

Question

Can a computer decide the truth of inequalities involvingexpressions built up from x and sin x?

NO! (Richardson 1968)

Idea: Let p, L ∈ Z[x1, . . . , xn] be such that L(~x)� p(~x)2.

f (~x) := −1 + 4p(~x)2 + L(~x)(sin2 πx1 + · · ·+ sin2 πxn).

If f (~x) < 0, then

sin2 πxi ≈ 0, so xi is very close to an integer ai , and

p(~x) < 1/2, which forces p(a1, . . . , an) = 0

Conclusion:

f < 0 somewhere ⇐⇒ p(~x) = 0 has an integer solution

(undecidable)


analysis

Bjorn Poonen

Group theory

F.p. groups

Word problem

Markov properties

Topology

Fundamental group


Manifold?

Knot theory

Analysis

Inequalities

Complex analysis

Integration

Inequalities in one variable

Question

Can a computer at least decide the truth of trigonometricinequalities in one variable?

NO! In fact, the one-variable inequality problem is just ashard as the many-variable inequality problem.

The proof uses the parametrized curve

~G (t) := (t sin t, t sin t3).

What does this curve in R2 look like?


analysis

Bjorn Poonen

Group theory

F.p. groups

Word problem

Markov properties

Topology

Fundamental group


Manifold?

Knot theory

Analysis

Inequalities

Complex analysis

Integration

As t ranges over real numbers,

~G (t) := (t sin t, t sin t3)

traces out

For a continuous function f (x , y),

f (x , y) ≥ 0 on R2 ⇐⇒ f ( ~G (t)) ≥ 0 for all t


analysis

Bjorn Poonen

Group theory

F.p. groups

Word problem

Markov properties

Topology

Fundamental group


Manifold?

Knot theory

Analysis

Inequalities

Complex analysis

Integration

Equality of functions

Bad news for automated homework checkers:

Theorem

It is impossible for a computer to decide,given two functions built out of x , sin x , | |,whether they are equal.

Proof: If you can’t decide whether f (x) ≥ 0, then you can’tdecide whether f (x) and |f (x)| are the same function.


analysis

Bjorn Poonen

Group theory

F.p. groups

Word problem

Markov properties

Topology

Fundamental group


Manifold?

Knot theory

Analysis

Inequalities

Complex analysis

Integration

Complex analysis

Example

Does

ez = w3 + 5z + 4

ew = w2 + 3z4 − 7

w4 = z9 + z5 + 2.

have a solution in complex numbers z and w?

Question

Can a computer decide whether a system of equationsinvolving the complex exponential function has a complexsolution?


analysis

Bjorn Poonen

Group theory

F.p. groups

Word problem

Markov properties

Topology

Fundamental group


Manifold?

Knot theory

Analysis

Inequalities

Complex analysis

Integration

Complex analysis

Question

Can a computer decide whether a system of equationsinvolving the complex exponential function has a complexsolution?

NO! (Adler 1969)Proof: The 3 steps below characterize Z in C by equations:

1. 2πiZ is the set of solutions to ez = 1

2. Q ={a

b: a, b ∈ 2πiZ and b 6= 0

}3. Z is the set of q ∈ Q such that 2q ∈ Q; thus

Z := {q ∈ Q : ∃z ∈ C such that ez = 2 and eqz ∈ Q}.

Thus

Hilbert’s tenth problem ⊆ the complex analysis problem.

Hilbert’s tenth problem is undecidable,so the complex analysis problem is undecidable.


analysis

Bjorn Poonen

Group theory

F.p. groups

Word problem

Markov properties

Topology

Fundamental group


Manifold?

Knot theory

Analysis

Inequalities

Complex analysis

Integration

Integration

Question

Can a computer, given an explicit function f (x),

1. decide whether there is a formula for∫f (x) dx ,

2. and if so, find it?

Theorem (Risch)

YES.

Theorem (Richardson)

NO.

Another answer: MAYBE; it’s not known yet.

All of these answers are correct!

undecidability in group theory, topology, and...

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